What is the integral of e^{(y^2)/2} ?

The integral of e^{(y^2)/2} is (pi/2)^{1/2}\erfi(y/(sqrt(2)))+C

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integral of e^{(y^2)/2}

integral

e^{\frac{y ^{2}}{2}}

(\frac{π}{2})^{\frac{1}{2}} erfi (\frac{y}{2}) + C

Solution steps

\int e^{\frac{y ^{2}}{2}} d y

Apply exponent rule:

\frac{a ^{n}}{b ^{n}} = (\frac{a}{b})^{n}

\frac{y ^{2}}{2} = (\frac{y}{2})^{2}

= \int e^{(\frac{y}{2})^{2}} d y

Apply u-substitution

: \int e^{u^{2}} 2 d u

= \int e^{u^{2}} 2 d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int e^{u^{2}} d u

Use the nonelementary integral:

\int e^{u^{2}} d u = \frac{π}{2} erfi (u)

= 2 \frac{π}{2} erfi (u)

Substitute back

u = \frac{y}{2}

= 2 \frac{π}{2} erfi (\frac{y}{2})

Simplify

2 \frac{π}{2} erfi (\frac{y}{2}) : (\frac{π}{2})^{\frac{1}{2}} erfi (\frac{y}{2})

= (\frac{π}{2})^{\frac{1}{2}} erfi (\frac{y}{2})

Add a constant to the solution

= (\frac{π}{2})^{\frac{1}{2}} erfi (\frac{y}{2}) + C

x (x^{2})

\frac{1}{( 1 - 2 x ^{2} )}

y = 4 t + 2

\frac{2}{( x ^{2} - 9 )}

\frac{1}{2 x + 1}

What is the integral of e^{(y^2)/2} ?
The integral of e^{(y^2)/2} is (pi/2)^{1/2}\erfi(y/(sqrt(2)))+C